60 A. Mukliopadhyay — HydroJcinetic Circulation. [No. 1, 



taken round the closed curve is clearly equal to the sum of the values of 



d\ dx 



ff 



taken round the projections of the closed curve on the coordinate planes. 

 Now, for the projected curve on the coordinate plane of yz, we have at 

 once from the ordinary formulae for the transformation of multiple 

 integrals, 



// 



=// 



dXdx 

 dX dx dX dx^ 



\ dy dz dz dy) "^ 



The projected curves on the other two coordinate planes lead to two 

 similar expressions. Hence, the circulation round the given closed 

 curve is furnished by 



I {udx + vdy + wdz) 



+ //(s|-||)-- 



^ j J \I.Ty-dyI^Y^''y- 

 But, as an immediate consequence of Clebsch's transformation, we have 



whence 



dx dx 



dy dy 



d<l> ^ dx 

 dz dz 



dy dz dy dz dz dy 



2 ^^_^_^^_dXdx 



dz dx ~ dz dx dx dz 



2^ = ~ _ ^ - ^ ^^ ^^^X 

 dx dy dx dy dydx' 



